Probability distributions of landslide volumes · Probability distributions of landslide volumes M. T. Brunetti, F. Guzzetti, and M. Rossi Consiglio Nazionale delle Ricerche, Istituto

Nonlin. Processes Geophys., 16, 179–188, 2009www.nonlin-processes-geophys.net/16/179/2009/© Author(s) 2009. This work is distributed underthe Creative Commons Attribution 3.0 License.

Nonlinear Processesin Geophysics

Probability distributions of landslide volumes

M. T. Brunetti, F. Guzzetti, and M. Rossi

Consiglio Nazionale delle Ricerche, Istituto di Ricerca per la Protezione Idrogeologica, Perugia, via Madonna Alta 126,06128 Perugia, Italy

Received: 28 August 2008 – Revised: 26 February 2009 – Accepted: 26 February 2009 – Published: 11 March 2009

Abstract. We examine 19 datasets with measurementsof landslide volume,VL, for sub-aerial, submarine, andextraterrestrial mass movements. Individual datasets in-clude from 17 to 1019 landslides of different types, includ-ing rock fall, rock slide, rock avalanche, soil slide, slide,and debris flow, with individual landslide volumes rang-ing over 10−4 m3

≤VL≤1013 m3. We determine the prob-ability density of landslide volumes,p(VL), using kerneldensity estimation. Each landslide dataset exhibits heavytailed (self-similar) behaviour for their frequency-size dis-tributions, p(VL) as a function ofVL, for failures exceed-ing different threshold volumes,V ∗

L , for each dataset. Thesenon-cumulative heavy-tailed distributions for each datasetare negative power-laws, with exponents 1.0≤β≤1.9, andaveragingβ≈1.3. The scaling behaviour ofVL for the en-semble of the 19 datasets is over 17 orders of magnitude,and is independent of lithological characteristics, morpho-logical settings, triggering mechanisms, length of period andextent of the area covered by the datasets, presence or lackof water in the failed materials, and magnitude of gravita-tional fields. We argue that the statistics of landslide volumeis conditioned primarily on the geometrical properties of theslope or rock mass where failures occur. Differences in thevalues of the scaling exponents reflect the primary landslidetypes, with rock falls exhibiting a smaller scaling exponent(1.1≤β≤1.4) than slides and soil slides (1.5≤β≤1.9). Weargue that the difference is a consequence of the disparity inthe mechanics of rock falls and slides.

1 Introduction

Landslides are caused by meteorological and geophysicaltriggers, and by human actions. Despite landslides beingfrequent and widespread, the statistics of landslide size are

Correspondence to:M. T. Brunetti([email protected])

not fully determined and remain poorly understood (Starkand Hovius, 2001; Malamud et al., 2004b; Van Den Eeck-haut et al., 2007). In this work, we exploit 19 datasets listingmeasurements of landslide size to determine the probabilitydensity of landslide volume,p(VL). Results provide empir-ical evidence that the non-cumulative distribution of land-slide volumes obeys a self-similar behaviour across 17 or-ders of magnitude. The finding has implications for landslidemodelling (e.g., Hergarten and Neugebauer, 1998; Katz andAharonov, 2006; Juanico et al., 2008), for landscape analysis(e.g., Hovius et al., 1997; Stark and Hovius, 2001; Malamudet al., 2004a; Korup, 2006; Guzzetti et al., 2008, 2009), forthe determination of landslide hazard (Guzzetti et al., 2005,2006), and as a starting point for the adoption of a quantita-tive scale of landslide magnitude, based onVL (Malamud etal., 2004b; Guzzetti et al., 2009).

We organize this paper by first giving a background to cu-mulative and non-cumulative frequency-size distribution for-mulas, and past studies that have examined the statistics oflandslide volumes (Sect. 2). We follow this be a descriptionof the 19 datasets used in this paper (Sect. 3), frequency-sizeanalyses of the data (Sect. 4), discussion of the implicationsof our results (Sect. 5), and finally our conclusions (Sect. 6).

2 Background

To determine the statistics of landslide sizes (i.e., area orvolume), investigators have used the cumulative or the non-cumulative number-size distributions. Inspection of the liter-ature (see below) reveals that, above a threshold, the cumu-lative distribution of large and very-large landslide volumesis generally well approximated by a negative power-law,

NL = rV −αL (1)

where NL is the cumulative number of landslides,VL

the landslide volume,α the cumulative power-law scal-ing exponent, andr a constant. The corresponding

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180 M. T. Brunetti et al.: Probability distributions of landslide volumes

Table 1. Characteristics of datasets in the literature listing measurements of landslide volumes,VL, for which the density distributions wereestimated.

Sourcea Geographic area Number of Distribution Distributionc Volume range Fit range Units Landslide Triggere

landslides typeb typed

1 Japan 800 c NL=105.49·V −0.85

L4×102–106 103–106 m3 m r

2 Iceland 224 c NL∝V −αL

1×106–4×107 m3 rs u3 New Zealand 42 c NL∝V −α

L1×106–5×108 m3 ra u

4, 5, 6 Alberta, Canada 409 c NLy∝V −0.72L

1×10−6–1×101 10−1–101 m3 rf u

6 Highway 99, British Columbia 390 c NLy=0.77·V −0.43L

1×10−2–8×104 1×100–8×104 m3 rf, rs m

6 BCR line, British Columbia 403 c NLy=0.12·V −0.40L

1×10−2–2×104 1×100–2×104 m3 rf, rs m

6 Highway 1, British Columbia 226 c NLy=1.36·V −0.70L

1×10−2–2×104 1×100–2×104 m3 rf, rs m

6 CP line, British Columbia 918 c NLy=1.13·V −0.65L

1×10−2–3×104 1×100–3×104 m3 rf, rs m

7 Hong Kong 2811 c NL=2.33·V −0.79L

1×10−2–1×105 3×100–1×105 m3 rf, m r

8 Grenoble, France 87 c NLy=4.2·V −0.41L

5×10−1–1×106 5×101–1×106 m3 rf m

8 Arly gorges, France 59 c NLy=8.5·V −0.45L

5×100–1×104 2×101–3×103 m3 rf m

8 Yosemite, USA 101 c NLy=4.5·V −0.46L

1×100–1×106 5×101–1×106 m3 rf m

9 World wide 142 c NLy∝V −0.52L

1×103–2×1010 3×107–2×1010 m3 rf u

9 Mahaval, La Reunion 370 c NLy∝V −1.0L

<9×106 m3 rf m10 Storega, off shore Norway ≈60 c NL=29.6–14.2·logVL 1×10−2–1×103 1×10−2–1×102 km3 sm u11 Queen Charlotte Islands, Canada 475 nc p(VL)∝V −1.87

L≈1×102

−1.6×104 5×102–1.6×104 m3 df, ds † r

11 Queen Charlotte Islands, Canada 140 nc p(VL)∝V −2.94L

≈1×102−1.2×103 2×102–1.2×103 m3 df, ds ‡ r

12 Yosemite, USA 392 nc f (VL)=10·V −1.1L

1×10−1–6×105 1×10−1–6×105 m3 rf, rs m

13 Umbria Marche, central Italy 155 nc f (VL)=0.1·V −1.2L

9.9×10−5–2.0×102 9.9×10−5–2.0×102 m3 rf e

13 Balza Tagliata, Umbria, Italy 62 nc f (VL)=0.1·V −1.2L

8.1×10−3–1.3×102 8.1×10−3–1.3×102 m3 rf e

13 Balza Tagliata, Umbria, Italy 1696 nc f (VL)=0.01·V −1.6L

2.4×10−5–5.4×101 2.4×10−5–5.4×101 m3 rf* e

14 Multiple areas nc f (VL)=2.34·V −1.07L

10−12–10−3 km3 rf m

15 Capilano watershed, Canada 2984 nc f (VL)∝V−βL

,1.1≤β≤3.6 1×101–2×10−5 m3 ss m

a 1, Fujii (1969); 2, Whalley et al. (1983); 3, Whitehouse and Griffiths (1983); 4, Gardner (1970); 5, Gardner (1983); 6, Hungr et al. (1999);7, Dai and Lee (2001); 8, Dussauge-Peisser et al. (2002); 9, Dussauge et al. (2003); 10, Issler et al. (2005); 11, Martin et al. (2002); 12,Guzzetti et al. (2003); 13, Guzzetti et al. (2004); 14, Malamud et al. (2004b); 15, Brardinoni and Church (2004).b c, cumulative distribution; nc, non-cumulative distribution.c NL, cumulative number;NLy , cumulative number or frequency per year;f (VL), frequency density;p(VL), probability density.d rf, rock fall; rs, rock slide; ra, rock avalanche; ss, soil slide; df, debris flow; sm, submarine; m, multiple; u, unknown; †, primary failure; ‡,sidewall failure; *, rock fragments.e r, rainfall; e, earthquake; m, multiple events or triggers; u, unknown.

non-cumulative number-volume distributionNL′ also fol-

lows a power-law,

NL′=

dNL

dVL

=−αrV−(α+1)L =sV

−βL ≡f (VL)≡NLT p(VL) (2)

with dNL the number of landslides in the volume interval[VL; VL+dVL] (i.e. in a “bin” of size dVL), β the non-cumulative scaling exponent, ands a constant. Equation (2)further shows that the non-cumulative number-volume dis-tribution, NL

′, is equivalent to the frequency density ofVL,f (VL). The corresponding probability density ofVL, p(VL),is obtained by dividingf (VL) by the total number of land-slides,NLT . For a non-cumulative power-law distributionwith exponentβ>1, the corresponding cumulative distri-bution, obtained by integration or summing, has exponentα=β–1 (Guzzetti et al., 2002).

We now discuss past literature that has examined thefrequency-volume distribution of landslides. A summary ofthese 15 studies is given in Table 1. In this literature re-view table, for each study, we have indicated geographic

area, number of landslides considered, whether the authorsused cumulative or non-cumulative frequency-volume statis-tics and over which range they were fitting the data, size dis-tribution found, the landslide type in their studies (rock fall,rock slide, rock avalanche, soil slide, slide, debris flow, sub-marine landslide) and the trigger of the landslides (rainfall,earthquake).

Fujii (1969) was probably the first to investigate the statis-tics of landslide size. Studying an inventory of 800 land-slides caused by heavy rainfall in Japan, he obtained a cu-mulative number-volume distribution that correlated with apower-law relation,α=0.85 (Eq. 2). Whalley et al. (1983)examined data for 224 large rockslides in Iceland. White-house and Griffiths (1983) studied 42 rock avalanche de-posits in the central Southern Alps of New Zealand. Thetwo datasets exhibited power-law scaling of the frequencyof VL, with a value forα not specifically given. Gard-ner (1983) investigated rock falls and rock slides in Alberta,Canada, and provided an exponential relationship for the fre-quency ofVL. Hungr et al. (1999) have given a cumulative

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M. T. Brunetti et al.: Probability distributions of landslide volumes 181

frequency-volume distribution for 1937 rock falls and rockslides in south-western British Columbia, divided into foursubareas, exhibiting negative power-laws withα=0.5±0.2.Dai and Lee (2001) examined 2811 landslides, mostly rockfalls, in Hong Kong in the period 1992–1997, and foundthat the cumulative frequency-volume distribution followeda negative power-law, withα=0.8. Dussauge-Peisser etal. (2002) compiled rock fall inventories for the Grenoblearea and for the Arly gorges, France, and determined thatthe cumulative frequency distributions of the rock fall vol-umes exhibited negative power-law behaviour withα=0.41and α=0.45, respectively. Dussauge et al. (2003) studieda world-wide inventory of 142 rock falls, and an inven-tory of 370 instrumental measurements of rock fall failuresat Mahaval, La Reunion, and established that the cumula-tive frequency-volume distributions were negative power-lawfunctions, withα=0.52 andα=1.0, respectively. Issler etal. (2005) studied submarine debris flows in the Storega land-slide area, off shore Norway, and found that the cumulativenumber-volume distribution was approximated by a logarith-mic relationship,NL=29.6−14.2 logVL, with VL in km3.

Martin et al. (2002) found that the non-cumulative prob-ability density for the volume of shallow landslides in theQueen Charlotte Islands, British Columbia, correlated witha negative power-law relation, withβ=1.87 for primary(open slope) failures, andβ=2.94 for gully (sidewall) fail-ures. Guzzetti et al. (2003) analyzed a catalogue of histor-ical rock falls in Yosemite National Park, USA, compiledby Wieczorek et al. (1992), and determined that the non-cumulative frequency density of rock fall volumes exhib-ited power-law scaling, withβ=1.1. Guzzetti et al. (2004)examined two inventories of earthquake induced rock fallsin the Umbria-Marche Apennines, Italy, and establishedthat the non-cumulative frequency density of rock fall vol-umes obeyed power-law scaling, withβ=1.2. Malamud etal. (2004b) re-examined the catalogues of rock falls availablefor the Yosemite area (Wieczorek et al., 1992), for the Greno-ble area (Dussauge-Peisser et al., 2002), and for the Umbriaregion (Guzzetti et al., 2004), and gave a non-cumulative fre-quency density distribution for all three data sets, over twelveorders of magnitude, that was well approximated by a nega-tive power-law, withβ=1.1. Brardinoni and Church (2004),measured the volume of shallow landslides in the Capi-lano watershed, British Columbia, and gave non-cumulativefrequency-volume relationships that followed power-law re-lations, withβ=2.7 to 3.6.

Review of the literature reveals that the proposed empiricaldistributions of landslide volumes obey almost invariably anegative power-law scaling (Table 1). Variability exists in theexponent of the power-law tails of the distributions. Part ofthe variability is natural i.e., due to different landslide typesand local morphological and lithological settings. However,part of the variability is fictitious, and caused by the differentmethods used to collect the data and to estimate the distribu-tions (White et al., 2008).

3 Available data

For our own analyses, we used 19 datasets available in theliterature with measurements of landslide volume,VL. Wesummarize the characteristics of each of these datasets in Ta-ble 2. Of the available datasets, 17 list the volumes of sub-aerial landslides (A to O, Q to R, in Table 2), one lists sub-marine failures (P), and one lists extraterrestrial mass move-ments (S). Multiple types of mass movements are considered,including rock fall, rock slide, rock avalanche, soil slide,slide, and debris flow (Cruden and Varnes, 1996). The num-ber of landslides in each dataset varies from 17 (O) to 1019(J), and the volume of the individual landslides spans 17 or-ders of magnitude, from 1.0×10−4 m3 (A) to 1.8×1013 m3

(S). The techniques used to measure individual landslide vol-umes varies. For some datasets (e.g., A, B, C, D, O) indi-vidual landslide volumes were measured in the field. Forother datasets (e.g., E, F, G, J, L, N) the landslide volumewas obtained by multiplying landslide length, width, anddepth measured in the field, on topographic maps, or fromaerial photographs (e.g., Simonett, 1967; Rice et al., 1969;Rice and Foggin, 1971; Larsen and Torres-Sanchez, 1996,1998; Baum et al., 2000; Barnard et al., 2001). For the otherdatasets, the technique used to measureVL is unknown.

Measurements of landslide volume are affected by vari-ous types of errors, which are difficult to quantify. For mostdatasets, the error associated with the measurement ofVL

is unknown. Based on geomorphological evidence and fieldexperience, Guzzetti et al. (2009) argued that the error asso-ciated with the measurement ofVL is within the same orderof magnitude as the measured volume. For this work, weadopted a similar approach, and we attributed an uncertaintyto the measurements ofVL in each dataset, based chiefly onthe landslide type and on information on the method used toobtainVL (Table 2). We assumed that the uncertainty de-pends primarily on the difficulty in estimating the landslidesub-surface geometry and depth. We divided the 19 datasetsin three groups, and attributed to each group a level of uncer-tainty, expressed as a rangeI of possible values for the true(unknown) landslide volume. Datasets listing landslide vol-umes measured in the field or using quantities (i.e., length,width, area) measured in the field or obtained through de-tailed analysis of aerial photographs, were attributed a rangeI=[0.5VL; 1.5VL], corresponding to an uncertainty of 50%.Datasets reporting submarine or extraterrestrial landslides,or landslides for which no information was available on themethod used to measure the landslide volume, were assigneda rangeI=[0.2VL; 5VL], corresponding to a factor of five un-certainty. Other datasets were attributed a rangeI=[0.5VL;2VL], corresponding to a factor of two uncertainty. We ac-knowledge that the adopted classification is heuristic andsomewhat subjective, but we maintain it provides a reason-able proxy for the error associated with the measurement ofVL.

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Table 2. Characteristics of 19 landslide datasets used in this study, listed in ascending order of minimum landslide volume. Last two columnsshow (i) the minimum landslide volumeV ∗

Lselected for the least-square linear fitting of the tail of the probability distribution, and (ii) the

corresponding scaling exponentβ of the negative power-law fit.

ID Sourcea Dataset Number of Range ofVL (m3) Ib Landslide Rock Triggere Age or periodf Threshold Slopelandslides,NLT min max min max typec typed V ∗

L(m3) β±εg

A 1 Umbria-Marche, central Italy 133 1.0×10−4 2.0×102 0.5VL 1.5VL rf, rs s e 29 Sep–14 Oct 1997 1.0×10−1 1.3±0.0B 1, 2 Balza Tagliata, Umbria, Italy 62 4.0×10−3 1.3×102 0.5VL 1.5VL rf s e 14 Oct 1997 1.0×10−1 1.4±0.0C 3, 4 Grenoble, France 105 3.0×10−2 5.0×108 0.5VL 1.5VL rf s m 1935 to 1995 1.0×101 1.2±0.0D 5 Yosemite, USA 379 5.0×10−2 1.1×107 0.5VL 1.5VL rf, rs i m 1867 to 2002 1.0×101 1.3±0.0E 6 New Guinea 207 3.4×10−1 1.4×106 0.5VL 2VL sl m m m 1.0×102 1.2±0.0F 7 Himalaya, India 338 5.0×10−1 4.0×105 0.5VL 2VL m m m m 1.0×102 1.5±0.0G 8, 9 Southern California, USA 66 8.0×10−1 8.5×102 0.5VL 1.5VL ss, df m r 1965 to 1969 3.2×101 1.6±0.1H 10 AVI archive, Italy 916 1.0×100 6.0×108 0.5VL 2VL m m m 1900 to 2000 1.0×103 1.2±0.0I 3, 4 Arly gorges, France 25 5.0×100 1.0×104 0.5VL 1.5VL rf s, p m 1954 to 1976 1.0×102 1.2±0.1J 11, 12 Puerto Rico, USA 1019 5.3×100 3.7×104 0.5VL 1.5VL ss, df v r ∼1950 to 1990 1.0×102 1.9±0.0K 13, 14 Debris flow 46 1.0×101 3.8×109 0.5VL 2VL df u r m 1.0×107 1.3±0.0L p.c. Tanaro valley, Northern Italy 100 2.4×101 8.4×103 0.5VL 1.5VL sl s r Nov 1994 3.2×102 1.8±0.1M 15 World slides 404 7.5×101 2.9×1010 0.2VL 5VL m m m m 1.0×107 1.5±0.1N 16 Washington, USA 20 7.6×101 1.1×106 0.5VL 2VL ss, df s r u 1.0×103 1.3±0.0O 17 Peat slides, Shetland Is., UK 17 2.3×103 5.9×104 0.5VL 1.5VL ss s m u 6.3×103 1.6±0.1P 18, 19 Submarine 43 6.0×103 5.0×1012 0.2VL 5VL sm s u u 1.0×1010 1.0±0.1Q 19, 20 Non volcanic 153 3.0×104 2.1×1010 0.2VL 5VL df, ra m r u 1.0×107 1.1±0.0R 19, 21 Volcanic 55 1.8×105 2.8×1010 0.2VL 5VL df, lh v m u 1.0×109 1.1±0.1S 19, 22 Mars-Moon 28 2.0×108 1.8×1013 0.2VL 5VL ra, rs u u u 1.0×1010 1.0±0.0

a 1, Antonini et al. (2002); 2, Guzzetti et al. (2004); 3, Dussauge-Peisser et al. (2002); 4, Dussauge et al. (2003); 5, Wieczorek et al. (1992);6, Simonett (1967); 7, Barnard et al. (2001); 8, Rice et al. (1969); 9, Rice and Foggin (1971); 10, Guzzetti and Tonelli (2004); 11, Larsenand Torres-Sanchez (1996); 12, Larsen and Torres-Sanchez (1998); 13, Iverson (1997); 14, Iverson et al. (1998); 15, Guzzetti et al. (2009);16, Baum et al. (2000); 17, Dykes and Warburton (2008); 18, Hampton et al. (1996); 19, Legros (2002); 20, Erismann and Abele (2001); 21,Hayashi and Self (1992); 22, McEwen (1989); p.c., P. Aleotti, personal communication to FG (2002).b uncertainty, expressed as a range of possible values for the true (unknown) landslide volume; see text for explanation.c rf, rock fall; rs, rock slide; ra, rock avalanche; ss, soil slide; sl, slide; df, debris flow; lh, lahar; sm, submarine; m, multiple; u, unknown.d i, intrusive; p, metamorphic; s, sedimentary; v, volcanic; m, multiple events or triggers; u, unknown.e r, rainfall; e, earthquake; m, multiple; u, unknown.f m, multiple dates or periods; u, unknown date or period.g see Fig. 4;ε=0.0 indicates that the errorε is smaller than 0.05, andε=0.1 indicates that the error is 0.05≤ε≤0.09.

No rigorous definition exists for the completeness of alandslide dataset (Malamud et al., 2004b; Guzzetti et al.,2009). In this work, a complete dataset is a dataset forwhich it is reasonable to assume that all landslides within therange of volume allowed by the adopted measuring techniquewere measured. Completeness of the 19 studied datasets de-pends on the measurement technique. Datasets listing land-slides caused by a single triggering event (Malamud et al.,2004b), or obtained through direct field survey (e.g., A, B,C, G, I, J, L, O), are considered, to a large extent, complete.The Yosemite dataset (D), obtained through a systematic andlong-term record of rock falls and rock slides covering sev-eral years (Wieczorek et al., 1992), is considered substan-tially complete for the period 1980–2002, and incompletefor the previous period (1867–1979) (Guzzetti et al., 2003).Other datasets (e.g., E, F, H, K, M, N, P, Q, R, S) are in-complete, and their degree of completeness remains undeter-mined.

4 Data analysis

We analyzed the 19 landslide datasets individually to get es-timates of the probability density of landslide volumep(VL),and to establish similarities and differences between theprobability distributions. To minimize problems related tothe use of different statistical methods for the estimation ofthe probability density ofVL, we selected a single methodand applied it to all the available datasets.

Inspection of the individual histograms obtained from theraw data reveals that the non-cumulative distributions ofVL

exhibit many more small values than large ones, and appear(even with some bins with no values) to decay as a heavy-tail. An example for Umbria, Italy (dataset B) is shown inFig. 1. Heavy-tailed data are commonly evaluated on log-arithmic scales (e.g., Newman, 2005; White et al., 2008).Hence, we transformed theVL measurements taking the log-arithm (base 10) ofVL, and obtained an ensemble of datasetswith −4.0≤logVL≤13.3. To estimate the probability densityof logVL, p(logVL), we adopted kernel density estimation(KDE) (Silverman, 1986; Scott, 1992; Venables and Ripley,2002). KDE replaces each value (in our case, each logVL



0 20 40 60 80 100 1200

10

20

30

40

50

1 10 1001

10

100

Landslide volume, VL (m3)

Cou

nt, n

Cou

nt, n


(a)

(b)

Fig. 1. Histograms show the number of landslides,n, as a func-tion of landslide volume,VL, for the Balza Tagliata dataset listingNLT =62 earthquake induced rock falls in Umbria, central Italy (Bin Table 2). Upper panel(a) shows count and volume in linear co-ordinates. Lower panel(b) shows count and volume in logarithmiccoordinates.

measurement) with an associated kernel function, character-ized by a given shape and bandwidth. We experimented withdifferent kernel functions (i.e., Gaussian, triangular, rect-angular) and bandwidths, and obtained optimally smoothedcurves using a Gaussian kernel, with bandwidth (standarddeviation) in the range between 0.3 and 0.5, depending onthe dataset. KDE was performed using the “density” func-tion in the R software for statistical computing, release 2.6.2(http://www.r-project.org/). The “density” algorithm dis-tributed the mass of the empirical distribution on a regulargrid of n=10 000 points, used a fast Fourier transform to con-volve the approximation with a discrete version of the kernel,

−5 0 5 10

Log of landslide volume, logVL(m3)

Pro

babi

lity

dens

ity, p

(log

VL)

(m-3)

A

B

C

D

DE

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

Z

15

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 2. Probability density of the logarithm of landslide volumep(logVL) as a function of logVL, shown in linear coordinates. Prob-ability densities were obtained through kernel density estimation.Letters A to S refer to dataset ID in Table 2. Letter Z refers to alldataset landslide volumes combined into one dataset (dashed blackline).

and evaluated the density at selected bins through linear ap-proximation (Silverman, 1986; Venables and Ripley, 2002).

Results of KDE are shown in Fig. 2 for the 19 considereddatasets (A–S), and for an additional dataset (Z, dashed blackline) that encompasses all the availableVL measurements(4116 data). The inspection of Fig. 2 reveals (i) that sev-eral of the shown probability density distributions, obtainedthrough KDE, are approaching symmetrical shapes, and (ii)that the probability density for the cumulative dataset Z hasconsiderably longer tails. The latter is the result of the lim-ited number of data for very small and very large landslides,i.e., for small and for large logVL values.

In Fig. 2, the individual probability densities,p(logVL),are smoother than the corresponding histograms, and satisfythe normalization condition:

n∑i=1

pi(logVL)[(logVL)i+1 − (logVL)i

]= 1, (3)

wheren=10 000, andpi(logVL) is the probability density forthei-th bin. Since we are interested in the probability densityof VL, p(VL), we transformedp(logVL). The probabilitydensityp(VL) has to satisfy the same normalization condi-tion:

n∑i=1

pi(VL)[(VL)i+1 − (VL)i

]= 1 (4)


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184 M. T. Brunetti et al.: Probability distributions of landslide volumesP

roba

bilit

y de

nsity

, p (V

L) (m

-3)

10−13

10−9

10−5

10−1

103

10−17


10-5 100 105 1010 1015

Log of landslide volume, logVL (m3)

-5 0 5 10 15

3

-1

-5

-9

-13

-17

Log

of p

roba

bilit

y de

nsity

,log(

p (V

L)) (m

-3) β = 1.3

AA

BZ

E

F

G

M

K

J

O

IL

N

D

H

C

Q

R

P S

Fig. 3. Probability densities of landslide volumes,p(VL), as afunction ofVL, shown in logarithmic coordinates (bottom and leftaxes). The same plot shows the logarithm of the probability densitylog(p(VL)) as a function of log(VL) in linear coordinates (top andright axes). Letters A to S refer to dataset ID in Table 2. Letter Zrefers to all dataset landslide volumes combined into one dataset.

Considering Eqs. (3) and (4), we obtained:

pi(VL) =pi(logVL)

[(logVL)i+1 − (logVL)i

](VL)i+1 − (VL)i

(5)

wherepi(VL) is the probability density ofVL for thei-th bin.Results of the transformation,p(VL) as a function ofVL, areshown in Fig. 3 using logarithmic coordinates.

In Fig. 3, we observe that for volumes in the range101 m3

≤VL≤109 m3, the probability densitiesp(VL) fordataset Z (i.e. all data combined) obeys the general trend out-lined by the ensemble of the probability density estimatesobtained for the 19 individual datasets. This general trendtends towards a negative power-law with exponentβ≈1.3.However, for very small (VL<101 m3) and for very large(VL>109 m3) landslides, the probability densities for datasetZ underestimates (i.e., are lower than) the densities obtainedfor individual datasets. This is a consequence of the shapeof the density estimation for dataset Z (see Fig. 2), whichis conditioned by the comparatively low number ofVL mea-surements for very small and very large landslides in the cu-mulative dataset.

To better quantify the differences in the scaling of the tailsof the probability densities, we obtained fits of the tails ofthe individual distributions, forVL greater than a thresh-old volumeV ∗

L (Table 2). V ∗

L was selected for each datasetthrough visual inspection. To account for problems associ-ated with the fitting of data spanning multiple orders of mag-

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100 105 1010 101510−5


Sca

ling

expo

nent

, β

2.0

1.8

1.6

1.4

1.2

1.0

0.8

SP

RQ

CE

I

K

H

M

B

DA

J

L

O

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N

Fig. 4. Results of 100 simulations for each individual landslidedataset A to S, that take into account the uncertainty on landslidevolumesVL (see Table 2, and text). For each individual dataset,a rectangle shows: (i) (x-axis) the range of landslide volumesVL,taking into consideration the rangeI of possible values for the true(unknown) landslide volume; (ii) (y-axis) mean of the scaling ex-ponentsβ of the power-law tail for the 100 simulations, with thevertical height of the rectangle representing the standard deviationof the β for those simulations. Colours show different landslidetypes: red for rock falls and rock slides, green for slides and soilslides, and white for other landslide types. Letters A to S refer todataset ID in Table 2.

nitude (e.g., the least square minimization criteria may notwork), VL and p(VL) were log-transformed (upper x-axisand right y-axis in Fig. 3). A straight line was fitted to thelog-transformed data, which is entirely equivalent to a nega-tive power-lawp(VL)∝V

−βL (Eq. 2).

To account for the estimated error associated with the mea-surement ofVL (Table 2), for each empirical dataset we con-structed 100 synthetic datasets, each obtained by substitut-ing the individual empirical measurements ofVL, with corre-sponding synthetic values,VS . Each syntheticVS value wasobtained by randomly sampling from a uniform distributionspanning a volume rangeI , determined by the correspond-ing VL measurement (Table 2). Power laws were fitted tothe individual synthetic datasets, and statistics for the scalingexponentsβ were determined (Table 2).

Figure 4 summarizes the results. For each landslidedataset, the range of landslide volumesVL, including the es-timated uncertainty, is shown along the x-axis. The corre-sponding variation of the scaling exponentβ for each dataset(i.e., one standard deviation centred on the mean value), isshown along the y-axis. For substantially complete datasets,β is related to the landslide type. Rock falls and rock slides(A, B, C, D, I) exhibit lower scaling exponents (1.1≤β≤1.4)



than slides and soil slides (G, J, L, O) (1.5≤β≤1.9). This isan indication that the relative proportion of large landslides islarger for rock falls, and smaller for slides. We attribute thedifference to a disparity in the mechanics of rock falls andslides (Malanud et al., 2004b).

5 Discussion

Inspection of the 19 non-cumulative probability distributionsof landslide volume obtained for the individual datasets re-veals significant similarities and some differences (Fig. 3).Each individual dataset exhibits a distinct self-similar be-haviour ofp(VL) as a function ofVL, for failures exceeding athreshold size,V ∗

L (Table 2). The tails of the distributions arereasonably well approximated by negative power-laws, with1.0≤β≤1.9, average ofβ≈1.3, median 1.3, and standard de-viation 0.3 (Table 2, Fig. 4). The scatter in the exponentsof the tails of the probability distributions is more limitedcompared to the spread of the scaling exponents reported inthe literature (Table 1). We attribute the reduced scatter tothe technique adopted to obtain the probability density ofVL

and to fit power-laws to the (heavy) tails of the distributions.Scatter in the value of the scaling exponents has multiple

reasons, including: (i) different landslide types, that reflectdifferent failure mechanisms (Cruden and Varnes, 1996),(ii) the natural variability inherent to landslide phenomena,and (iii) the number, quality, and completeness of the sin-gle datasets (Stark and Hovius, 2001; Guzzetti et al., 2002;Malamud et al., 2004b; Van Den Eeckhaut et al., 2007). Thescaling of the power-law fits are independent of time, andof the period covered by the individual datasets that range(where known) from tens of seconds for earthquake-inducedlandslides (∼101 s, B), to a few hours for rainfall-inducedlandslides (∼104 s, L), to multiple decades (∼109 s, C, D, G,H, I, J) (Table 2).

Some dataset (e.g., A, G, N, P) exhibit a deviation of thetail from the power-law behaviour, for the largest landslidesin each dataset. For these datasets, the power-law fit over-estimates the probability density, i.e. the empirical density islower than the density modelled by the power-law fit. Thebehaviour has been observed for other natural hazards, in-cluding earthquakes (Turcotte, 1997). For landslides, it canbe the result of: (i) under-sampling of the largest events e.g.,because of the lack of data, or because the temporal span ofa dataset is small compared to the frequency of large (rare)events, and (ii) geometrical constrains, e.g., a landslide can-not be larger than the slope where the failure occurs (Guzzettiet al., 2002).

A few datasets (e.g., F, G, J, L) exhibit a “rollover” ofthe probability densities for smallVL. This is similar to the“rollover” identified in the frequency and probability den-sity functions of landslide areas (e.g., Pelletier et al., 1997;Stark and Hovius, 2001; Guzzetti et al., 2002; Brardinoni andChurch, 2004; Malamud et al., 2004b; Van Den Eeckhaut et

al., 2007). The “rollover” can be the result of under-samplingof the smallest landslides (Stark and Hovius, 2001), or a geo-morphological (physical) property of the landslides, markinga change in the dominant processes that contribute to land-slide formation and initiation (Guzzetti et al., 2002, 2008;Katz and Aharonov, 2006; Malamud et al., 2004b). Datasetsshowing a rollover contain landslides predominantly of theslide type, whereas rock falls do not exhibit a rollover. Weattribute the difference to different failure mechanisms forthe two landslide types (Malamud et al., 2004b).

Small datasets withn≤55 (I, K, N, O, P, R, S) show multi-ple deviations of the probability density from the power-lawfit. We attribute the irregularities in the density functions forthese datasets to the reduced size of the datasets, and to (par-tial) failure of the kernel density estimation to capture thelandslide density.

In Fig. 3, the position of the individual probability densityestimates is influenced by: (i) the range ofVL, that controlsthe placing along the x-axis, (ii) the scaling of the distribu-tion, which is similar for most of the datasets, and (iii) thenormalization condition, expressed by Eq. (4). Location ofthe distributions along the x-axis or the y-axis is not influ-enced by the number of measurements in the datasets. In-spection of Fig. 3 reveals that the ensemble of the densityfunctions clusters around a trending line that follows a neg-ative power-law with a scaling exponentβ≈1.3, across 17orders of magnitude, 10−4 m3

≤VL≤1013 m3. The findingindicates a self-similar (fractal) (Turcotte, 1997; Turcotte etal., 2002; Cello and Malamud, 2006) behaviour ofp(VL) asa function ofVL, over a very large range of volumes. Thisis in spite of lithological characteristics, morphological set-tings, triggering mechanisms, length of period and extent ofthe area covered by the datasets, presence or lack of water inthe failed soils and rocks, and intensity of gravitational fieldsfor extraterrestrial landslides (S) (Table 2). The scaling be-haviour is the result of natural processes that control land-slide volumes, and not an artefact introduced by the tech-nique used to collect the samples, by the methods used tomeasure landslide volumes, or by the number of data in theindividual datasets.

Further inspection of Fig. 3 reveals that the scaling of thedistributions covering small (e.g., A, B), or medium (e.g.,I, J), or large (e.g., P, S) landslides is rather similar. Thisis an indication that landslides of all sizes (in the range10−4 m3

≤VL≤1013 m3) belong essentially to the same un-derlying frequency-size distribution. In other words, forlandslide volumes the largest (“extreme”) failures pertain tothe same distribution as the other landslides. This finding isimportant because it allows using a range ofVL for whichdatasets are most abundant, of higher quality, or more com-plete, to infer the scaling behaviour for other ranges ofVL

for which data are limited, and of poor or of uncertain qual-ity. The fact that the 19 datasets individually cover only areduced portion of the entire range ofVL measurements is anindication that single datasets are bounded by geometrical or



geomorphological limits. As discussed above, the reducednumber of small landslides in a dataset is the result of a sam-pling problem, or a change in the process that initiates thelandslides (Pelletier et al., 1997; Guzzetti et al., 2002; Mala-mud et al., 2004b; Katz and Aharonov, 2006). Conversely,the largest landslides in a dataset provide indication on thelargest slope failure that can be caused by an individual trig-ger of given magnitude in an area (e.g., B in Table 2), orthe largest slope failure that can form in a given landscape(Guzzetti et al., 2002, 2008).

We suggest that the distinct self-similar scaling behaviourof landslide volumes across several orders of magnitude isconditioned primarily on the geometrical properties of thesystem (i.e., slope or rock mass), where slope failures occur,and subordinately on the environmental (e.g., morphological,lithological, geological, meteorological, climatic) conditionsthat characterize the system. In other words, the statistics ofVL is controlled by the geometrical scaling (spacing, length,persistence) of the discontinuities (e.g., bedding, lamination,schistosity, fracture, cleavage, joint, fault) that control thelocation and initial volume of the slope failures (Katz andAharonov, 2006). This is confirmed by the fact that rock fallsand rock slides have lower scaling exponents (1.1≤β≤1.4)than slides and soil slides (1.5≤β≤1.9), a consequence ofthe different mechanics of the two types of mass movements(Cruden and Varnes, 1996).

Our finding represents an advancement in the understand-ing of the statistics of landslides (Stark and Hovius, 2001;Malamud et al., 2004b; Van Den Eeckhaut et al., 2007) thatcan be used to confront physical (Katz and Aharonov, 2006)and computer (Hergarten and Neugebauer, 1998, Juanico etal., 2008) models of landslide sizes. The scale-invariant be-haviour of landslide volume can prove useful to model ero-sion rates and sediment fluxes in landscapes dominated bymass-wasting processes (Hovius et al., 1997; Stark and Hov-ius, 2001; Malamud et al., 2004a; Korup, 2006; Guzzettiet al., 2008, 2009), where more detailed information is notavailable. The statistics ofVL are also important for land-slide hazard assessment (Guzzetti et al., 2005, 2006), andspecifically for modelling rock fall hazard (Guzzetti et al.,2003, 2004). Lastly, we anticipate our results to contributeto establish a physically-based magnitude scale for landslideevents, based on landslide volume (Malamud et al., 2004b;Guzzetti et al., 2009).

6 Conclusions

Through kernel density estimation, we obtained the probabil-ity densities of landslide volumes for 19 datasets, which in-cluded sub-aerial, submarine and extraterrestrial mass move-ments. Analysis of the probability densities,p(VL), indi-cates that each dataset exhibits negative power-law scaling ofp(VL) as a function ofVL, for landslides exceeding a thresh-old volume. The ensemble of the probability density esti-

mates obtained for the individual datasets follows a negativepower-law (scale invariant trend) with exponentβ≈1.3, inthe range 10−4 m3

≤VL≤1013 m3. This behaviour ofp(VL)

over 17 orders of magnitude is in spite of local geomorpho-logic settings, triggering mechanisms, length of period andextent of the area covered by the datasets, presence or lackof water in the failed soils and rocks, and intensity of grav-itational fields. We suggest that the self-similar behaviourof landslide volumes is controlled primarily by the geome-try and abundance of the discontinuities in the soil and rockmass where landslides initiate. Differences in the scaling ex-ponent of the density distributions reflect differences in thelandslide types, with rock falls systematically exhibiting asteeper tail (1.5≤β≤1.9) than slides (1.1≤β≤1.4). The dif-ference reflects a disparity in the mechanical behaviour ofrock falls and slides.

Our results advance the understanding of the statistics oflandslide volumes, and can contribute to the validation ofphysical (Katz and Aharonov, 2006) and computer (Her-garten and Neugebauer, 1998; Juanico et al., 2008) mod-els of landslide sizes, to regional studies of landslide hazard(Guzzetti et al., 2003, 2004, 2005, 2006), and to the mod-elling of erosion and sediment fluxes in landslide dominatedlandscapes (Hovius et al., 1997; Stark and Hovius, 2001;Malamud et al., 2004b; Korup, 2006; Guzzetti et al., 2008,2009). The research is also a starting point to establish aphysically-based scale for landslide events, based on land-slide volume (Malamud et al., 2004a; Guzzetti et al., 2009).

Appendix A

The KDE script

To determine the probability density of landslide vol-ume, p(VL), the author MTB prepared a script for theR free software environment for statistical computing,release 2.6.2 (http://www.r-project.org/). The follow-ing are available for download through the journalWeb site, and at the universal resource locator addresshttp://geomorphology.irpi.cnr.it/tools/stat-ls/pdf-ls-vol:(i) the script KDE landslidevolumes.r, (ii) two ex-ample datasets Data volumem3 UmbriaMarche.txt,Data volumem3 BalzaTagliata.txt, (iii) a file with illustra-tive resultsOutput.pdf. The two demonstrative datasets areprovided as text files, and list datasets A and B in Table 2.The script can be customized to analyze different landslidevolume data.

Acknowledgements.We are grateful to the colleagues that haveprovided individual landslide datasets. We thank two reviewers, fortheir constructive comments that improved the text substantially.Research supported by European Commission Project 12975(NEST) Extreme Events: Causes and Consequences (E2-C2).MTB and MR were supported by grants provided by the Italian


http://www.r-project.org/

http://geomorphology.irpi.cnr.it/tools/stat-ls/pdf-ls-vol


Space Agency (ASI MORFEO) and the Italian National Depart-ment for Civil Protection.

Edited by: B. MalamudReviewed by: two anonymous referees

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Probability distributions of landslide volumes · Probability distributions of landslide volumes M. T. Brunetti, F. Guzzetti, and M. Rossi Consiglio Nazionale delle Ricerche, Istituto